Equations and Inequalities

Complex Numbers and Imaginary Numbers

The set of complex numbers consists of all numbers in the form a + bi, where a and b are real numbers, and i is the imaginary unit. The imaginary unit i is defined as the number whose square is -1.

• Imaginary Unit:

• Standard Form: Every complex number can be written as

Operations on Complex Numbers

Complex numbers can be added, subtracted, and multiplied using the same rules as binomials. The operations are performed component-wise for addition and subtraction, and using the distributive property for multiplication.

• Addition/Subtraction:

• Multiplication:

Example: Adding and Subtracting Complex Numbers

Example: Multiplying Complex Numbers

• (since )

Conjugate of a Complex Number

The conjugate of a complex number is . Multiplying a complex number by its conjugate results in a real number:

Complex Number Division

To divide complex numbers, multiply the numerator and denominator by the conjugate of the denominator to eliminate the imaginary part from the denominator.

• For , multiply numerator and denominator by :

The denominator simplifies to , a real number.

Example: Using Complex Conjugates to Divide Complex Numbers

• Multiply numerator and denominator by :

Principal Square Root of a Negative Number

The principal square root of a negative number (where ) is defined as:

Example: Operations Involving Square Roots of Negative Numbers